Journal of Applied Geophysics 123 (2015) 71–80

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Journal of Applied Geophysics

j ourna l homepage: www.e lsev ie r .com/ locate / j appgeo

Improved estimation of P-wave velocity, S-wave velocity, andattenuation factor by iterative structural joint inversion of crosswellseismic data

Tieyuan Zhu a,⁎, Jerry M. Harris b

a The University of Texas at Austin, Jackson School of Geosciences, Austin, TX 78712, USAb Stanford University, Department of Geophysics, Stanford, CA, 94305, USA

⁎ Corresponding author.E-mail addresses: tzhu@jsg.utexas.edu (T. Zhu), jerry.h

(J.M. Harris).

http://dx.doi.org/10.1016/j.jappgeo.2015.09.0050926-9851/© 2015 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 14 October 2014Received in revised form 20 August 2015Accepted 4 September 2015Available online xxxx

Keywords:Joint inversionCross well seismicReservoir characterization

We present an iterative joint inversion approach for improving the consistence of estimated P-wave velocity, S-wave velocity and attenuation factormodels. This type of inversion scheme links two ormore independent inver-sions using a joint constraint, which is constructed by the cross-gradient function in this paper. The primary ad-vantages of this joint inversion strategy are: avoiding weighting for different datasets in conventionalsimultaneous joint inversion,flexible for incorporating prior information, and relatively easy to code.Wedemon-strate the algorithmwith two synthetic examples and two field datasets. The inversions for P- and S-wave veloc-ity are based on ray traveltime tomography. The results of thefirst synthetic example show that the iterative jointinversion take advantages of both P- and S-wave sensitivity to resolve their ambiguities aswell as improve struc-tural similarity between P- and S-wave velocity models. In the second synthetic and field examples, joint inver-sion of P- and S-wave traveltimes results in an improved Vs velocity model that shows better structuralcorrelation with the Vp model. More importantly, the resultant VP/VS ratio map has fewer artifacts and is bettercorrelated for use in geological interpretation than the independent inversions. The second field example illus-trates that the flexible joint inversion algorithm using frequency-shift data gives a structurally improved attenu-ation factor map constrained by a prior VP tomogram.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

Due to data deficiency and complex characteristic of geological sys-tem, e.g. multiple fluids in the reservoir, single geophysicalmodelmightbe difficult to characterize the geological target fully. For example, com-pressional waves are very sensitive to gas-saturated rocks while shearwaves are not. Attenuation is potentially more sensitive than velocityto the amount of gas in a rock (Winkler and Murphy, 1995). The ratioof Vp/Vs is more sensitive to changes of fluid type than Vp or Vs sepa-rately (Dvorkin et al., 1999; Hamada, 2004). It is also believed that at-tenuation factor is closely related to permeability (Pride et al., 2003).We can see that different geophysical models tend to reflect comple-mentary characters of reservoir. It is natural to combine several typesof geophysical data collected over the same reservoir region to reduceambiguity in inversion results, leading tomore reliablemodels for reser-voir characterization.

A type of joint inversion refers to combining several different typesof geophysical datasets in a single inversion algorithm and then simul-taneously or iteratively solving a least-squares problem (Vozoff and

arris@stanford.edu

Jupp, 1975; Haber and Oldenburg, 1997; Julia et al., 2000; Gallardoand Meju, 2003). Simultaneous joint inversion approaches have beensuccessfully applied for different geophysical data to provide improvedgeophysical models (e.g., Gallardo and Meju, 2004; De Stefano, 2007;Linde et al., 2008; Doetsch et al., 2010; De Stefano et al., 2011; Gao etal., 2012; Lelievre et al., 2012). However, coupling two or more datasetsin a single inversion still face some difficulties, especially large-scaleproblem: first, the huge coupled Jacobian and/or Hessian matrices forthe different data inversions have to be computed and/or stored for si-multaneous use (Hu et al., 2009); second, the determination of suitablerelative weighting between different objective functions can be chal-lenging (Gallardo and Meju, 2007; Moorkamp et al., 2011).

In this paper, we discuss an alternative approach to simultaneousjoint inversion for the tomography problem that is quite similar to theones of Hu et al. (2009) and Heincke et al. (2010). The iterative joint in-version couples independent inversions through iterationswith a cross-constraint term. At every iteration, we still run an independent inver-sion byminimizing an objective functionwith the additional cross-con-straint term. The presented approach overcomes the memory issue andthe determination of relative weighting of different data sets. The crossconstraint could be a direct parameter relation or a structural link. A di-rect parameter relation for different models based on the empirical orrock-physics relations (e.g., Carcione et al., 2007) may be limited in

Fig. 1. Flowchart of iterative joint inversion scheme. The boxes ‘A’ and ‘B’ represent the in-dependent inversions. The box ‘J’ represents the joint constraint term between ‘A’ and ‘B’.

72 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

some specific places. Instead, we implement a structural link—the cross-gradient function—which measures the structural similarity betweenthe different models instead of the direct parameter relation (Gallardoand Meju, 2003; Zhu and Harris, 2011; Um et al., 2014).

Another advantage of the iterative joint inversion algorithmwith thecross-gradient structural constraint is its flexibility to incorporate priormodels into the independent algorithms. For example, a prior lithologicmap (e.g., from reflection-migrated images) could be applied to con-strain other parameters in the cross-gradient function. This is easy toimplement as an additional regularization term in an independent in-versionwithin a single unknown, but it is difficult to use formultiple un-knowns, whose unknownsmay not be on the same order of magnitude(e.g., velocity and resistivity).

We begin by presenting a flexible iterative joint inversion frame-work that allows us to test different geophysical datasets. We thengive an overview of the different parts of the joint inversion framework:the objective function definition, the cross-gradient function, and thedetermination of regularization weights. We test the algorithm withtwo synthetic examples for jointly inverting Vp and Vs models. Finally,we apply the approach to two seismic cross-well field datasets acquiredat the west Texas for reservoir characterization.

2. Methodology

The inverse problem is formulated as an optimization that mini-mizes an objective functionΦ, which combines ameasure of datamisfit,Φd, a regularization measure Φm:

minΦ mð Þ ¼ Φd mð Þ þ λΦm mð Þ; ð1Þ

where the model vector m is a spatial function m(x,y,z), and λ is aregularization parameter, which is used to adjust contributions fordata misfit from the model regularization term and the constraintfunction.

The objective functions of the iterative joint inversion of twodatasets with a cross constraint ψcc(m1,m2)are defined as

Φ1 ¼ Φd m1ð Þ þ λ1Φm m1ð Þ þ β1ψcc m1;m2ð Þ;Φ2 ¼ Φd m2ð Þ þ λ2Φm m2ð Þ þ β2ψcc m1;m2ð Þ; ð2Þ

where m1 and m2 denote two models for two correspondingdatasets. In such an iterative joint inversion, we still run two inversionsseparately. In each independent inversion, the cross constraint is func-tional as a new regularization and includes complementary informationfrom a jointmodel during iterations. The coefficient β controls the influ-ence from other models on the solution through the cross constraint.Through the constraint ψcc(m1,m2), two independent inversions in Eq.(2) exchange information (e.g., geologic structure) during iterations.

The data misfit Φd and the regularization terms Φm are written as

Φd mið Þ ¼ Wd G mið Þ−dobsi

� ���� ���L2; ð3Þ

Φm mð Þ ¼ 12

Wmk k22 ð4Þ

where ‖⋅‖22 represents an L2-norm, and all quantities written in boldrepresent vectors. The subscript i refers to the index of multiple models(dataset), G(m) is the forward functional, dobs is the observed data vec-tor, andm is the unknown model vector.Wd is the data weighting ma-trix, which ensure the data by giving appropriate weights in theinversion (see Eq. (15) in Pidlisecky et al., 2007). The regularizationterm W is chosen as the first- and second-order spatial derivatives(Zhu and Harris, 2015). A finite-difference approximation of the W in3D results in the sparse matrix

W ¼ axGx þ ayGy þ azGz þ alapL ð5Þ

where ax,ay and az are relatively weights applied to x ,y, and z spatialcomponents of the discrete gradient (Gx,Gy,Gz) (Pidlisecky et al., 2007),L is the discretized Laplacian matrix (Aster et al., 2005), and alap is theweighting value.

For our problem, the cross-gradient function is chosen as the con-straint functionalψcc=ψcg(m1,m2). The constraint functional is definedas ψcg(m1,m2)=‖t‖22, where the cross-gradient function t is defined inGallardo and Meju (2003):

t x; y; zð Þ ¼ ∇m1 x; y; zð Þ � ∇m2 x; y; zð Þ; ð6Þ

where ∇ is the gradient in the x,y and z directions. The structuralsimilarity requires t = 0, which means that any spatial changes occur-ring in both m1 and m2 must point in the same or opposite direction,or no spatial changes in one of m1 and m2 (Gallardo and Meju, 2004).The derivatives of the cross-gradient termwith respect to themodel pa-rameters are given in 2D (Gallardo andMeju, 2004) and 3D (Tryggvasonand Linde, 2006). The Jacobian matrix Jxgis then obtained. Each row ofJacobian matrix has six nonzero elements of 2Nm (Nm is the modelsize) (cf. Gallardo and Meju, 2004, Eq. (9)).

In our synthetic and field examples, we carefully choose λ throughseveral tests to balancemodel misfit and data misfit in the independentinversion. When λ is obtained, we use this value for the iterative jointinversion. We determine the β value by the experienced rule given byHu et al. (2009)

β ¼ 10L Φmj j2= N ψccj j2 þ δ2� �h i

; ð7Þ

whereN=NxNyNz and δ is a small value. L usually ranges 0bLb5 anddepends on which model is superior, i.e., the superior model has rela-tively small weights. Nx, Ny and Nz are the number of discretized gridpoints in the x,y and z directions.

Fig. 1 shows the flowchart of our iterative procedure. The procedurebegins with two input datasets and their corresponding initial modelsm0=(m1,m2),which are usually homogenous in our tomography algo-rithm. In the first iterative, we run two independent inversions (box ‘A’and ‘B’) for m1

1 and m21. The superscript denotes the iteration number.

When we obtained updated models m11 and m2

1 from flows ‘A’ and ‘B’

73T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

respectively, the cross constraint box ‘J’ is updated fromupdatedmodelsm1

1 andm21 for the next iteration. Then we have the new objective func-

tions with new constraint functions for each independent inversion in‘A’ and ‘B’. The procedure is repeated iteratively until the predefined it-eration number or the data error tolerances for both independent inver-sions are satisfied. The output inversion results are the optimal solutionsmoptimal=(m1,m2).

Fig. 2. a) True P-wave velocity model, b) S-wave velocity model, c) Vp/Vs ratio model, and d) cpendent inversions. i)–l): Corresponding inverted models by iterative joint inversion.

All following examples come from crosswell seismic tomography.The forward modeling of refraction traveltimes is used to solve theeikonal equation by the finite-difference method (Vidale, 1990; Zeltand Barton, 1998). Times are calculated away from a source on thesides of an “expanding” cube, one side being completed before thenext is considered. We use a Gauss–Newton strategy to solve the inde-pendent inverse problem. Computation details of gradient and Hessian

ross-gradient map between two models. e) - h): Corresponding inverted models by inde-

Fig. 3. Crossplot of VP and VS obtained from independent inversion, joint inversion, andtrue model. (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

Table 1Final model misfit, data misfit, and RMS of cross-gradient functions for synthetic dataset I.

Inversions Normalized RMSof model misfit

Normalized RMSof data misfit

RMS of cross-gradient value

P-wave S-wave P-wave S-wave

Independent 0.036 0.037 0.011 0.012 0.25Joint 0.033 0.034 0.010 0.010 0.09

74 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

of the regularization term can be found in Zhu and Harris (2015). Thelinearized Hessian of the cross-gradient term is approximated by JxgT Jxg.

3. Synthetic examples

We first test the algorithm on a reservoir model in the crosswell ge-ometry. Fig. 2a and b show P- and S-wave velocity models. A gas–watersaturated reservoir is embedded in the second layer. Note that the P-wave velocity contrast between gas and water saturated zone is highat 24%. The relatively small differences in S-wave velocities of the gas-saturated sand and the water-saturated sand is only 6%, which makesit hard to identify the gas-water contact in the reconstructed S-wave ve-locity model. The joint inversion of P and S data is expected to resolvethis ambiguity.

In the first synthetic example, the model dimension is 500 m by1210 m. We set up the source-receiver geometry with 29 shots in theright well with a spacing of 40 m and 116 receivers in the left wellwith a spacing of 10 m. Thewell distance is about 450m.We calculatedP- and S-wave traveltime data from the true models in Fig. 2a and b bysolving the eikonal equation by the finite-difference method (Vidale,1990). No noise is added. The starting models for P- and S-wave inver-sions are homogeneous, with mean values of the true models. The reg-ularization parameters λ are 2×10−5 and 3×10−5 for the P- and S-wave model inversion algorithms, respectively. We ran ten iterationsfor both inversions, which is sufficient for convergence of the Gauss-Newton method.

Fig. 2e–h show inverted P- and S-wave velocity models by indepen-dent inversions, Vp/Vs ratio model, and its cross-gradient values. Theinverted P-wavemodel is quite good, especially at the gas-water contactbecause of high velocity contrast. But the lateral geometry of the objectsis not well recovered. The reason probably is limited ray aperture in thecrosswell geometry. Fig. 2f shows the inverted S-wave velocity model.The geometry of the gas-water reservoir zone is better defined becauserays fairly pass through this zone. However, the gas-water contact is notdelineated. We can see this model is difficult for either the P-wave or S-wavemethod alone to resolve. The right panel (Fig. 2h) shows the struc-tural similarity (cross-gradient) of the inverted P-wave velocity (e) andS-wave velocity models (f).

Next, we ran the iterative joint inversion algorithm for the P- and S-wavemodels. The same regularization λ are used as in the independentinversions. Fig. 2i–l show the joint inversion results. Overall, we can seethat the joint inversion results tend to remove artifacts seen in indepen-dent inversion results. Notably, the gas-water contact in the inverted S-wave velocity model (Fig. 2j) is better resolved and the edge of the res-ervoir in the P-wave velocity model (Fig. 2i) is better delineated. Below50 m, there are improvements in S-wave velocity structure but P-wavevelocity along the dipping channel become slightly smoother. Fig. 2kdisplays the Vp/Vs ratio model. The cross-gradient values from joint in-version (Fig. 2l) are closer to zeros, as designed. The root mean square

(RMS) value (defined as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑NxNz

i¼1t2i =NxNz

s� 1e6, where t(x,y,z) is defined

at Eq. (6)), at the final step decreases to ~0.09 from ~0.25 for indepen-dent inversion results. This implies that the joint inversion with thecross-gradient constraint produces structurally similar P- and S-wavemodels.

Fig. 3 presents the cross-plot of the P- and S-wave velocities. Thecross-plot values from true P- and S-wave velocity are shown in sevenred crosses that represent seven different blocks. The blue circles are in-dependent inversion values, while the yellow triangles indicate joint in-version results. The P and S velocity values of the joint inversionmodelsare somewhat less dispersed than those obtained from the independentinversion results. All model and data misfits are listed in Table 1. Thedata misfit between the observed data dobs and the calculated data dcalis defined as ‖dcal−dobs‖2/‖dobs‖2. The normalized RMS model misfit is‖mest−mtrue‖2/‖mtrue‖2. Both model and data misfits are decreasing to

slightly smaller values by joint inversion than by independentinversions.

In the second example, we test the algorithm in a more realisticmodel taken fromwest Texaswell logs. The VP log and its correspondingvelocity model are shown in Fig. 4a. The S-wave and density logs can befound in van Schaack et al. (1995).We set up the source-receiver geom-etry of a west Texas field dataset (Harris et al., 1995) with 201 shots inthe right well with a spacing of 2.5 m and 201 receivers in the left wellwith a spacing of 2.5 m. The source function is a Ricker wavelet with acentral frequency of 800 Hz. We use an elastic finite-difference solver(Wu et al., 2005) to generate the synthetic data set. Fig. 4b shows a com-mon shot gather at depth 227.5 m. The P- and S-wave picks are easilyidentified in Fig. 4b. We manually picked P- and S-wave traveltimes.Note that S-wave picks are not pickable in the near offset, due to thesource radiation pattern (Harris et al., 1995). The regularization param-eter λ = 2×10−5 is chosen for independent P- and S-wave inversions.

Fig. 5a–d show the synthetic P- and S-wave velocity models, VP/VS

ratio model, and their cross-gradient values that are zeros. The corre-sponding models obtained from independent inversions of these newdata are shown in Fig. 5e–h. The VP model recovers many of the struc-tural features of the testmodel but the bottom edge is not well resolved(Fig. 5e). However, the VS model is smooth, without much detail (Fig.5f). The S-wave model shows a curved high-velocity layer betweendepths about 100m to 200m, possibly because of incomplete small-off-set data. This curved interface may come from the ray bending in thispart of the model and the overlying low-velocity zone above depth100 m becomes poorly resolved. The normalized RMS model misfitsare 5.5% for inverted Vp model (Fig. 5e) and 5.8% for inverted Vsmodel (Fig. 5f). Fig. 5g shows the resultant VP/VS ratio map with strongartifacts. The independently inverted P- and S-wave velocity modelsshow non-similar structures, with an RMS value of 555.6 (Fig. 5f).

Thenwe ran the joint inversion of the P- and S-wave traveltime datawith cross-gradient constraints. We used the same starting Vp and Vsmodels as well as the same regularization λ as the independent inver-sions. Fig. 5i shows Vp model inverted by joint inversion that visuallysimilar to Fig. 5e. Note that the bottom layer is resulted correctly inFig. 5i. But it somehow appears curved layers, which might be causedby curvedVs layers. It reminds us that less confidenceVsmodelmayde-grade the Vp model through joint constraints. The normalized RMSmodel misfit is 4.8% for Vpmodel (see Table 2). Through joint inversion,the VS model (Fig. 5j) shows large improvements in comparison withthe independent inversion. For example, the S-wave low-velocity

Fig. 4. (a) P-wave velocity model interpreted from the sonic log (black line). Black triangles denote receivers while red stars represent sources. (b) Shot gather at 227.5 m depth. P-wavesare easily picked and constitute a complete dataset. S-wave picks are relatively easy tomake at the far offset but aremore difficult in the near offset, so the S-wave traveltime pick dataset isincomplete.

75T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

layer above depth 100m is better resolved and the high-velocity layer isflat. The jointly inverted models give the vertical stratification betterthan do the separately inverted models. The normalized RMS modelmisfit is 4.5% for inverted Vs model (Fig. 5j). Fig. 5k shows the resultantVP/VS ratio map with fewer artifacts and appears flatter for use in geo-logical interpretation than the map from the independent inversions.The RMS of the cross-gradient values is 31.9. The cross-gradient maps(Fig. 5h and l) imply that Vp and Vs models from the joint inversionaremore structurally similar than they do from independent inversions.The VP–VS cross-plots obtained from the independent and joint invertedmodels are shown in Fig. 6. Similar to the previous example, the esti-mated values are less scattered from the jointly inverted models thanthe independently inverted models.

4. Field examples

4.1. Joint inversion of VP and VS in the McElroy field

We now use the joint inversion algorithm to characterize Vp and Vsmodels using seismic P- and S-wave traveltime data collected betweenwells in west Texas. The baseline data recorded in 1993 before CO2 in-jection is chosen for this study, since it has relatively high-quality S-wave picks. The source-receiver system in McElroy field (Harris et al.,1995) consists 241 shots in the right well with a spacing of 2.5 ft. and241 receivers in the left well with a spacing of 2.5 ft. The real source-receiver geometry is in 3D with the maximum deviation of 20 ft. inthe Y direction (perpendicular to X-Z plane). Compared to the size ofX (horizontal) and Z (depth) directions, we simplify the tomographyproblem in an approximate 2D geometry (see Fig. 7a). The piezoelectricsource is used. The profile contains approximately 58,081 traces. Pre-processing of the crosswell data included manual traveltime pickingand correction of the source-receiver positions in the deviated bore-holes. Seismic P-wave traveltimes range between 8.9 and 15.4 ms, andS-wave traveltimes range between 14.7 and 34.5 ms, with estimatedpicking errors for both data sets of less than 1 ms. The original P- andS-wave traveltime data are shown in Fig. 6 in Harris et al. (1995). Dueto radiation pattern, S-wave energy is missing in the near offset butpickable in the far offset, shown in Fig. 7b, which is same as the second

synthetic example. So we limit the effective S-wave picks with inci-dence angles larger than 45° for inversion in this paper. The S-wavetraveltimedata contains 40,722 picks compared to 58,081 P-wave picks.

The 2D inverse domain size is 88 × 246, for the total number of21,648 unknowns. The grid spacings in the horizontal and vertical direc-tions are 2.5 ft. From the previous studies (Harris et al., 1995; Lazaratosand Marion, 1997), we know that the lithology is quite flat, so we used10 times more horizontal constraint than vertical (see Eq. (5)). The reg-ularization parameter λ = 1×10−5 for P- and S-wave models is keptconstant during the independent and joint inversion procedures. Thecoefficient of the cross-gradient term β is computed by setting L = 1in the VP inversion and L=4 in the VS inversion (see Eq. (7)), implyinglarger constraint in the VS inversion than the VP inversion using thiscross-gradient term.

Fig. 8a–d show the results of independent inversions for VP, VS,VP/VS, and the resulting cross-gradient models, respectively. The P-and S-velocity models show a high-velocity layer between 2750 ft(838 m) and 2850 ft (868 m) and possibly another between 2600(792 m) and 2700 ft (823 m). However, the S-velocity model shownless geologically shaped features. The VP/VS ratios (Fig. 8c) are difficultto use for geological prediction, as they have lots of variables, some ofwhich are artifacts. Fig. 8e–h show the results of the joint inversion,for which the VP and VS models correlate better. The high-velocitylayer between depths 2750–2850 ft is more pronounced in the S-ve-locity model (Fig. 8f). Interestingly, the resultant VP/VS ratio maphas fewer artifacts and appears flatter for use in geological interpre-tation than the map from the independent inversions. Fig. 8h showsthe improvement in the structural similarity between VP and VS im-ages as judged by the computed RMS value of the cross-gradientfunction (RMS = 89.3), which is smaller than that of the separatelyinverted models (RMS = 309.3). The data misfits are shown inTable 3. Joint inversion leads to the anomalously increased data mis-fit of P-wave traveltime data than that of independent inversion.There are possible reasons: using a 2D geometry in our inversioncould increase nonlinearity of this tomography with realistic 3D ge-ometry. This is true for both independent and joint inversion. But theweighting parameter on the joint structural constraint is likely not toeffectively reduce data misfit but makes inversion focus on enforcing

Fig. 5. Thefirst row: synthetic VPmodel (a), VSmodel (b), VP/VS ratio (c), and cross-gradient values betweenVP and VS (d). The second row: invertedVPmodel (e), VSmodel (f), VP/VS ratio(g), and cross-gradient values between VP and VS (h) reconstructed by independent inversions of P- and S-wave traveltime data. The third row: inverted VP model (i), VS model (j), VP/VS

ratio (k), and cross-gradient values between VP and VS (l) reconstructed by iterative joint inversion of P- and S-wave traveltime data.

76 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

structural similarity. Or, this similar-structure assumption is nottruly satisfied in the studying area.

Fig. 9 shows the scatter plot of VP–VS values from the invertedmodels and the sonic logs. The VP-VS trend from the joint inversion isless scattered and shows better agreement with that from sonic logs(blue). Similar observations using simultaneous joint inversion of Vpand Vs against independent inversions are also observed using localearthquake data by Tryggvason and Linde (2006).

Table 2Final modelmisfit, datamisfit, and RMS of cross-gradient functions for synthetic dataset II.

Inversions Normalized RMSof model misfit

Normalized RMSof data misfit

RMS of cross-gradient value

P-wave S-wave P-wave S-wave

Independent 0.055 0.058 0.009 0.011 555.6Joint 0.048 0.045 0.006 0.010 31.9

4.2. Constrained inversion of attenuation factor in the King Mountain site

In the secondfield example,we setup an inversion of the attenuationfactor constrained by using the VP cross-gradient constraint. The differ-ence from the first example is that we fix the VP model during iterativejoint inversion. The prior VP map is well obtained from the previousstudies (Langan et al., 1997; Zhu and Harris, 2015).

The crosswell data were collected for 201 sources spaced at approx-imately 1.5m (5 ft) depth intervals and203 receivers, also at 1.5m spac-ing. Thus, we have approximately 40,000 traces of raw data. The 2Dinverse domain size is 21× 206, for the total number of 4326unknowns.The grid spacings in the horizontal and vertical directions are 9.5 and1.5 m (32 and 5 ft), respectively.

We estimate the frequency-independent attenuation factor from thefirst arrivals in the crosswell field data collected in the King Mountainsite inwest Texas. The attenuation factor estimation is based on the cen-troid frequency-shift method (Quan and Harris, 1997). The centroid

Fig. 6. Crossplot of VP and VS obtained from independent inversion, joint inversion, andtrue model.

77T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

frequency-shift method is attractive for crosswell data because it is lesssensitive to wave behaviors that affect amplitudes and has a broad fre-quency band. The idea is simple. The centroid frequency-shift of thewavelet between the source position and the receiver position is pro-portional to the integral of the attenuation factor over the path. There-fore, the input data for inversion is the centroid frequency shiftbetween the reference wavelet and each trace. In practice, we can

Fig. 7. Source-receiver crosswell geometry (a) and a common shot gather data with theshot location at 2840 ft. (b). We can see that P-wave traveltimes are easily picked whileS-wave traveltimes are not complete due to source radiation pattern, missing in thenear offset.

measure the receiver centroid frequency from recorded seismograms,butmay not directly measure the source centroid frequency and its var-iance. For simplicity (to obtain relative attenuation), the source centroidfrequency is chosen as the maximum value of the receiver centroid fre-quency at the receivers. The source variance is chosen as the average ofthe receiver variance at the receivers (Quan and Harris, 1997). For theray-based approach, the ray pathmatrix is determined from velocity in-version.We took the velocity traveltime tomography code (see Zhu andHarris, 2015) using centroid frequency shift data rather than traveltimefor estimating the attenuation factor.

We performed two inversions: independent and joint. Two inver-sions stop at the tenth iteration. The starting models for both attenua-tion inversions are homogeneous. Fig. 10a shows the independent VP

model taken from Fig. 5 of Zhu andHarris (2015). Fig. 10b shows the at-tenuation factor model estimated by independent inversion. The atten-uation factor tomogram shows lateral heterogeneities in the reservoirdepth from 8700 to 9000 ft (~2651 m to 2712 m). However, the reser-voir zone may be hard to delineate from Fig. 10b. Moreover, the inde-pendent VP and attenuation factor tomograms give inconsistentgeological information about this area. The high-velocity layer betweendepths 8400 ft (2560 m) and 8500 ft (2590 m) is almost horizontal,while the corresponding low attenuation layer is slightly dipping.

In the implementation of the constrained joint inversion, since thevelocity model is fairly good, we put a large weight on the cross-gradi-ent term in the objective function of attenuation factor inversion. Fig.10c shows the resulting jointly estimated attenuation factor tomogram,which gives a consistent (structurally similar) image of the velocitymodel, especially the carbonate reservoir body. The final data misfitand cross-gradient values are shown in Table 4. Integrating velocityand attenuation factor results, we see that the reservoir zone exhibitsthe strongest attenuation, and corresponds to the low velocity, around16,000 ft./s (4876 m/s). The shale layers (between depths 8500 ft(2590 m) and 8700 ft (2651 m) and between depths 9100 ft(2773 m) and 9200 ft (2804 m)) exhibit the proportional relation be-tween velocity and attenuation, which means that regions with low/medium velocity correspond to medium/low attenuation. We remarkthe fact that the present iterative joint inversion algorithmwith a flexi-ble constraint would make attenuation factor model to be structurallysimilar to VP model, which might be more geologically interpretable.

5. Discussion

The first synthetic example shows that the presented joint inversionof P- and S-wave traveltime data apparently improves the P-wave andS-wave velocity models (See Fig. 1). In the case, the ambiguity in reser-voir geometry from P-wave velocity inversion is caused by limited raypath through reservoir. The reservoir geometry is well solved in the S-wave velocity inversion. On the other hand, the ambiguity of S-wave ve-locity inversion is caused by relativeweak contrast between the gas-sat-urated sand and the water-saturated sand while P-wave velocitybetween two sands exhibits high contrast. Therefore, taking advantagesof two models in the iterative joint inversion of P- and S-wavetraveltimes data can give improved P- and S-wave velocity models bymitigating their ambiguities.

However, this is not the case in the rest of examples. Either S-wavetraveltimes in the specific crosswell geometry are incomplete or fre-quency-shift attenuation data is with uncertainty when using the firstarrivals' waveform that is always interfered by later arrivals. Twomodels inverted from such an ‘imperfect’ dataset are believed to havelarger uncertainties than P-wavemodel inverted from complete and ro-bust P-wave first arrival traveltimes. In this situation, the iterative jointinversion takes efforts to improve the low-confident S-wave velocityand attenuation factor models by incorporating P-wave velocity modelbut not much improvement in the high-confident P-wave velocitymodel.

Fig. 8. Inverted models by independent (a–d) and joint inversions (e–h). a)–d): Inverted P-wave, S-wave velocity model, VP/VS and cross-gradient values between two models by inde-pendent inversion. e)–h): Corresponding results by joint inversion.

78 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

Although iterative joint inversion scheme avoid weighting betweenmultiple datasets in single objective function, it still requires theweighting between the data, the regularization, and the cross-gradientfunctional (i.e., λ,β), which adjust the tradeoff among structural similar-ity, model misfit, and data misfit. In this study, we specify the sameweights for the cross-gradient approach as independent inversions forcomparisons, which may impact the joint inversion results. For exam-ple, the joint inversion results show improved models with similar-

Table 3Final data misfit and RMS of cross-gradient functions for field dataset I.

Inversions Normalized RMS of P-wave traveltime data N

Independent 0.088 0.Joint 0.104 0.

structure, but P-wave model seems not to be improved consistently(e.g., see the dipping channel in Fig. 2i in the first synthetic exampleand P-wave model in Fig. 5i in the second synthetic example). Thismight be due to the un-optimal weights for the regularization termand cross-gradient constrained term, which reminds us to investigatethe weightings for the joint inversion approach in the future.

It is worthy to note that structural constraint using the cross-gradi-ent function is effective in our synthetic examples, provided that

ormalized RMS of S-wave traveltime data RMS of cross-gradient value

064 309.3059 89.3

Fig. 9. Crossplot of VP and VS obtained from independent inversion (red), joint inversion(yellow), and the sonic log (blue). The trend indicated by yellow dots is closer to thesonic log. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Table 4Final data misfit and RMS of cross-gradient functions for field dataset II.

Inversions Normalized RMS of frequency-shiftdata

RMS of cross gradientvalue

Independent 0.096 291Joint 0.095 248

79T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

structural similarity exists between physical parameters. When it vio-lates, the convergence of joint inversion with structural constraintmay be slow or even fail. If this is the case, other constraints, e.g.,petrophysical and empirical relation, may be the choice. Although we

Fig. 10. Inverted attenuation factor models by independent (b) and joint inversions (c). Inverte

show our iterative joint inversion algorithm with the cross-gradientstructural constraint, other constraints can also be incorporated intothis iterative joint inversion framework.

6. Conclusions

We have reported an iterative joint inversion approach for invertingP-wave velocity, S-wave velocity and attenuation factor models using astructural constraint. It is different from simultaneous joint inversion.Simultaneous joint inversion couples independent inversions in a singleiterative domain with a joint cross constraint. Careful attention must betaken for the regularization coefficients and relative weights betweenmultiple models for convergence. The iterative joint inversion frame-work avoids this selection of relative weights for different datasetsand flexibly allows us to change the model parameterization and num-ber of objective functions in order to investigate different coupling ap-proaches. It is also flexible to incorporate a prior model as a structuralconstraint in our iterative joint inversion. In addition, the iterative

d P-velocity model (a) is used as a cross-gradient constraint in the iterative joint inversion.

80 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) 71–80

joint inversion is very easy to code without many modifications in theoriginal independent inversion. This inversion framework is straightfor-wardly extended for inverting multiple datasets. The joint constraintcan also use other constraints, e.g., petrophysical and empiricalrelations.

We demonstrate the algorithm's feasibility using two synthetic ex-amples and two different field datasets from west Texas. Our resultsdemonstrate thebenefits of using iterative joint inversion are: (1) bettersimilarity in the geologic structural features between Vp and Vs modelsand between Vp and attenuation factor models; (2) moderate improve-ment in estimated values, e.g., improved VP-Vs relationships comparedwith those determined by well logs; (3) flexibility for constraining alower-confidence model with the use of a higher-confidence model. Inaddition, we found that inappropriate weighting parameters of thejoint constraint in our algorithm may degrade the high-confidencemodel. The future study should focus on the investigation of theweightings for the joint inversion approach.

Acknowledgments

Wewould like to thankDr. Youli Quan formanyhelp on tomographycode and many useful discussions on joint inversion when we startedthis research at Stanford. Critical comments by one anonymous review-er improved the paper. Tieyuan Zhu is supported by Jackson Distin-guished Postdoctoral Fellowship at the University of Texas at Austin.

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